Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review

Solutions of the Density Matrix-Pairing Tensor Equations of Superconductivity Theory

W. H. Young

Published Jul 1, 1963• 1 citation
Publisher

Abstract

An algorithm is provided for writing down explicitly all solutions of the density matrix-pairing tensor equations which arise in the generalized Bogoliubov-Valatin transformation approach to superconductivity theory. Certain simpler special cases are then examined. Finally reasons are given indicating that our solutions should provide a practical computational tool in many-body theory.

DOI
10.1103/physrev.131.476

Related papers

A density-matrix formulation of the hartree-fock-bogoliubov equations for space-dependent superconductivity

William L. Clinton

The Hartree-Fock-Bogoliubov equations that govern the quasi-particle states of space-dependent superconductivity are formulated in terms of the pure state density matrices ρn(′, )  Φn() Φ (). If is the Hartree-Fock-Bogoliubov Hamiltonian where HH is the normal metal part and Δ() the pairing potential, then the density operator equations are The pairing potential Δ() = V Σnun() V() tanh (βEE/2) is a mixed state quantity and, just as in the normal metal, the equations must be solved self-consistently. We use a steepest descent (optimum gradient) technique, developed by McWeeny for Hartree-Fock equations, to derive an iterative algorithm for solving the above equations. The formulism is developed in general for nonzero temperatures. The method is applied to the T ≈︁ TT region.

International Journal of Quantum Chemistry•Jun 18, 2009•5 citations

Reduced density matrix functional theory for superconductors

Jonathan Schmidt, Carlos L. Benavides-Riveros, Miguel A. L. Marques

We present an ab initio theory for superconductors, based on a unique mapping between the statistical density operator at equilibrium, on the one hand, and the corresponding one-body reduced density matrix $\ensuremath{\gamma}$ and the anomalous density $\ensuremath{\chi}$, on the other. This formalism for superconductivity yields the existence of a universal functional ${\mathfrak{F}}_{\ensuremath{\beta}}[\ensuremath{\gamma},\ensuremath{\chi}]$ for the superconductor ground state, whose unique properties we derive. We then prove the existence of a Kohn-Sham system at finite temperature and derive the corresponding Bogoliubov--de Gennes--type single-particle equations. By adapting the decoupling approximation from density functional theory for superconductors we bring these equations into a computationally feasible form. Finally, we use the existence of the Kohn-Sham system to extend the Sham-Schl\"uter connection and derive a first exchange-correlation functional for our theory. This reduced density matrix functional theory for superconductors has the potential of overcoming some of the shortcomings and fundamental limitations of density functional theory of superconductivity.

arXiv:1903.01516•Physical review. B./Physical review. B•Jun 5, 2019•27 citations

Self-Consistency in the Quasi-Chemical Equilibrium Theory of Superconductivity

Colin J. Thompson, Takeo Matsubara

By expressing the density matrix, for the special case of one pair state only and simple pairing, in exponential form, we reproduce the results of earlier papers and obtain an expression for the entropy which is formally identical with that of Bardeen, Cooper and Schrieffer and that of Bogoliubov. The expression for the entropy is shown to be exact in the limit of infinite volume. A variational calculation is then performed and the resulting set of equations is identical with that of B.C.S., thus establishing the relation between the Quasi-Chemical Equilibrium, the B.C.S., and the Bogoliubov theories of superconductivity at non-zero temperature.

Progress of Theoretical Physics•Apr 1, 1963•1 citation

Density Matrix Functional Theory with account of pairing correlations

S. Krewald, V. B. Soubbotin, V. Tselyaev +1 more

The extension of the density functional theory (DFT) to include pairing correlations without formal violation of the particle-number conservation condition is described. This version of the theory can be considered as a foundation of the application of existing DFT plus pairing approaches to atoms, molecules, ultracooled and magnetically trapped atomic Fermi gases, and atomic nuclei where the number of particles is exactly conserved. The connection with the Hartree-Fock-Bogoliubov theory is discussed. The method of the quasilocal reduction of the nonlocal theory is described. This quasilocal reduction allows to obtain equations of motion which are much more simple for the numerical solution than the equations corresponding to the nonlocal case.

arXiv:nucl-th/0412018•Dec 6, 2004•2 citations

Numerical Simulation Solution of the BCS Pairing Problem

Xufeng, Feng, Wang +5 more

We propose a new simulation computational method to solve the reduced BCS Hamiltonian based on spin analogy and submatrix diagonalization. Then we further apply this method to solve superconducting energy gap and the results are well consistent with those obtained by Bogoliubov transformation method. The exponential problem of 2N-dimensional matrix is reduced to the polynomial problem of N-dimensional matrix. It is essential to validate this method on a real quantum computer and is helpful to understanding the many-body quantum theory.

理论物理通讯：英文版•Jan 1, 2005•0 citations

Numerical Simulation Solution of the BCS Pairing Problem

Feng Xu, An-Min Wang, Xiao-Dong Yang +1 more

We propose a new simulation computational method to solve the reduced BCS Hamiltonian based on spin analogy and submatrix diagonalization. Then we further apply this method to solve superconducting energy gap and the results are well consistent with those obtained by Bogoliubov transformation method. The exponential problem of 2 N -dimensional matrix is reduced to the polynomial problem of N -dimensional matrix. It is essential to validate this method on a real quantum computer and is helpful to understanding the many-body quantum theory.

arXiv:quant-ph/0407100•Communications in Theoretical Physics•Jul 1, 2005•4 citations
An algorithm is provided for writing down explicitly all solutions of the density matrix-pairing tensor equations which arise in the generalized Bogoliubov-Valatin transformation approach to superconductivity theory. Certain simpler special cases are then examined. Finally reasons are given indicating that our solutions should provide a practical computational tool in many-body theory.
The Hartree-Fock-Bogoliubov equations that govern the quasi-particle states of space-dependent superconductivity are formulated in terms of the pure state density matrices ρn(′, )  Φn() Φ (). If is the Hartree-Fock-Bogoliubov Hamiltonian where HH is the normal metal part and Δ() the pairing potential, then the density operator equations are The pairing potential Δ() = V Σnun() V() tanh (βEE/2) is a mixed state quantity and, just as in the normal metal, the equations must be solved self-consistently. We use a steepest descent (optimum gradient) technique, developed by McWeeny for Hartree-Fock equations, to derive an iterative algorithm for solving the above equations. The formulism is developed in general for nonzero temperatures. The method is applied to the T ≈︁ TT region.
We present an ab initio theory for superconductors, based on a unique mapping between the statistical density operator at equilibrium, on the one hand, and the corresponding one-body reduced density matrix $\ensuremath{\gamma}$ and the anomalous density $\ensuremath{\chi}$, on the other. This formalism for superconductivity yields the existence of a universal functional ${\mathfrak{F}}_{\ensuremath{\beta}}[\ensuremath{\gamma},\ensuremath{\chi}]$ for the superconductor ground state, whose unique properties we derive. We then prove the existence of a Kohn-Sham system at finite temperature and derive the corresponding Bogoliubov--de Gennes--type single-particle equations. By adapting the decoupling approximation from density functional theory for superconductors we bring these equations into a computationally feasible form. Finally, we use the existence of the Kohn-Sham system to extend the Sham-Schl\"uter connection and derive a first exchange-correlation functional for our theory. This reduced density matrix functional theory for superconductors has the potential of overcoming some of the shortcomings and fundamental limitations of density functional theory of superconductivity.
By expressing the density matrix, for the special case of one pair state only and simple pairing, in exponential form, we reproduce the results of earlier papers and obtain an expression for the entropy which is formally identical with that of Bardeen, Cooper and Schrieffer and that of Bogoliubov. The expression for the entropy is shown to be exact in the limit of infinite volume. A variational calculation is then performed and the resulting set of equations is identical with that of B.C.S., thus establishing the relation between the Quasi-Chemical Equilibrium, the B.C.S., and the Bogoliubov theories of superconductivity at non-zero temperature.
The extension of the density functional theory (DFT) to include pairing correlations without formal violation of the particle-number conservation condition is described. This version of the theory can be considered as a foundation of the application of existing DFT plus pairing approaches to atoms, molecules, ultracooled and magnetically trapped atomic Fermi gases, and atomic nuclei where the number of particles is exactly conserved. The connection with the Hartree-Fock-Bogoliubov theory is discussed. The method of the quasilocal reduction of the nonlocal theory is described. This quasilocal reduction allows to obtain equations of motion which are much more simple for the numerical solution than the equations corresponding to the nonlocal case.
We propose a new simulation computational method to solve the reduced BCS Hamiltonian based on spin analogy and submatrix diagonalization. Then we further apply this method to solve superconducting energy gap and the results are well consistent with those obtained by Bogoliubov transformation method. The exponential problem of 2N-dimensional matrix is reduced to the polynomial problem of N-dimensional matrix. It is essential to validate this method on a real quantum computer and is helpful to understanding the many-body quantum theory.
We propose a new simulation computational method to solve the reduced BCS Hamiltonian based on spin analogy and submatrix diagonalization. Then we further apply this method to solve superconducting energy gap and the results are well consistent with those obtained by Bogoliubov transformation method. The exponential problem of 2 N -dimensional matrix is reduced to the polynomial problem of N -dimensional matrix. It is essential to validate this method on a real quantum computer and is helpful to understanding the many-body quantum theory.